66.2.7 problem Problem 7

Internal problem ID [13827]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER. Problems page 172
Problem number : Problem 7
Date solved : Wednesday, March 05, 2025 at 10:18:39 PM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+\frac {2 {y^{\prime }}^{2}}{1-y}&=0 \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)+2/(1-y(x))*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{1} x +c_{2} -1}{c_{1} x +c_{2}} \]

✓ Mathematica. Time used: 0.196 (sec). Leaf size: 37

ode=D[y[x],{x,2}]+2/(1-y[x])*D[y[x],x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {c_1 x-1+c_2 c_1}{c_1 (x+c_2)} \\ y(x)\to 1 \\ y(x)\to \text {Indeterminate} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)**2/(1 - y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(2)*sqrt((y(x) - 1)*Derivative(y(x), (x, 2)))/2 + Derivative(y(x), x) cannot be solved by the factorable group method